Fit an MCP- or SCAD-penalized regression path

Fit coefficients paths for MCP- or SCAD-penalized regression models over a grid of values for the regularization parameter lambda. Fits linear and logistic regression models, with option for an additional L2 penalty.

ncvreg(
  X,
  y,
  family = c("gaussian", "binomial", "poisson"),
  penalty = c("MCP", "SCAD", "lasso"),
  gamma = switch(penalty, SCAD = 3.7, 3),
  alpha = 1,
  lambda.min = ifelse(n > p, 0.001, 0.05),
  nlambda = 100,
  lambda,
  eps = 1e-04,
  max.iter = 10000,
  convex = TRUE,
  dfmax = p + 1,
  penalty.factor = rep(1, ncol(X)),
  warn = TRUE,
  returnX,
  ...
)

Arguments

X: The design matrix, without an intercept. ncvreg standardizes the data and includes an intercept by default.
y: The response vector.
family: Either "gaussian", "binomial", or "poisson", depending on the response.
penalty: The penalty to be applied to the model. Either "MCP" (the default), "SCAD", or "lasso".
gamma: The tuning parameter of the MCP/SCAD penalty (see details). Default is 3 for MCP and 3.7 for SCAD.
alpha: Tuning parameter for the Mnet estimator which controls the relative contributions from the MCP/SCAD penalty and the ridge, or L2 penalty. alpha=1 is equivalent to MCP/SCAD penalty, while alpha=0 would be equivalent to ridge regression. However, alpha=0 is not supported; alpha may be arbitrarily small, but not exactly 0.
lambda.min: The smallest value for lambda, as a fraction of lambda.max. Default is 0.001 if the number of observations is larger than the number of covariates and .05 otherwise.
nlambda: The number of lambda values. Default is 100.
lambda: A user-specified sequence of lambda values. By default, a sequence of values of length nlambda is computed, equally spaced on the log scale.
eps: Convergence threshhold. The algorithm iterates until the RMSD for the change in linear predictors for each coefficient is less than eps. Default is 1e-4.
max.iter: Maximum number of iterations (total across entire path). Default is 10000.
convex: Calculate index for which objective function ceases to be locally convex? Default is TRUE.
dfmax: Upper bound for the number of nonzero coefficients. Default is no upper bound. However, for large data sets, computational burden may be heavy for models with a large number of nonzero coefficients.
penalty.factor: A multiplicative factor for the penalty applied to each coefficient. If supplied, penalty.factor must be a numeric vector of length equal to the number of columns of X. The purpose of penalty.factor is to apply differential penalization if some coefficients are thought to be more likely than others to be in the model. In particular, penalty.factor can be 0, in which case the coefficient is always in the model without shrinkage.
warn: Return warning messages for failures to converge and model saturation? Default is TRUE.
returnX: Return the standardized design matrix along with the fit? By default, this option is turned on if X is under 100 MB, but turned off for larger matrices to preserve memory. Note that certain methods, such as summary.ncvreg() require access to the design matrix and may not be able to run if returnX=FALSE.
...: Not used.

Value

An object with S3 class "ncvreg" containing:

beta: The fitted matrix of coefficients. The number of rows is equal to the number of coefficients, and the number of columns is equal to nlambda.
iter: A vector of length nlambda containing the number of iterations until convergence at each value of lambda.
lambda: The sequence of regularization parameter values in the path.
penalty, family, gamma, alpha, penalty.factor: Same as above.
convex.min: The last index for which the objective function is locally convex. The smallest value of lambda for which the objective function is convex is therefore lambda[convex.min], with corresponding coefficients beta[,convex.min].
loss: A vector containing the deviance (i.e., the loss) at each value of lambda. Note that for gaussian models, the loss is simply the residual sum of squares.
n: Sample size.

Additionally, if returnX=TRUE, the object will also contain

X: The standardized design matrix.
y: The response, centered if family='gaussian'.

Details

The sequence of models indexed by the regularization parameter lambda is fit using a coordinate descent algorithm. For logistic regression models, some care is taken to avoid model saturation; the algorithm may exit early in this setting. The objective function is defined to be $$Q(\beta|X, y) = \frac{1}{n} L(\beta|X, y) + P_\lambda(\beta),$$ where the loss function L is the deviance (-2 times the log likelihood) for the specified outcome distribution (gaussian/binomial/poisson). See here for more details.

This algorithm is stable, very efficient, and generally converges quite rapidly to the solution. For GLMs, adaptive rescaling is used.

References

Breheny P and Huang J. (2011) Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection. Annals of Applied Statistics, 5: 232-253. doi:10.1214/10-AOAS388

Examples

# Linear regression --------------------------------------------------
data(Prostate)
X <- Prostate$X
y <- Prostate$y

op <- par(mfrow=c(2,2))
fit <- ncvreg(X, y)
plot(fit, main=expression(paste(gamma,"=",3)))
fit <- ncvreg(X, y, gamma=10)
plot(fit, main=expression(paste(gamma,"=",10)))
fit <- ncvreg(X, y, gamma=1.5)
plot(fit, main=expression(paste(gamma,"=",1.5)))
fit <- ncvreg(X, y, penalty="SCAD")
plot(fit, main=expression(paste("SCAD, ",gamma,"=",3)))

par(op)

op <- par(mfrow=c(2,2))
fit <- ncvreg(X, y)
plot(fit, main=expression(paste(alpha,"=",1)))
fit <- ncvreg(X, y, alpha=0.9)
plot(fit, main=expression(paste(alpha,"=",0.9)))
fit <- ncvreg(X, y, alpha=0.5)
plot(fit, main=expression(paste(alpha,"=",0.5)))
fit <- ncvreg(X, y, alpha=0.1)
plot(fit, main=expression(paste(alpha,"=",0.1)))

par(op)

op <- par(mfrow=c(2,2))
fit <- ncvreg(X, y)
plot(mfdr(fit))             # Independence approximation
plot(mfdr(fit), type="EF")  # Independence approximation
perm.fit <- perm.ncvreg(X, y)
plot(perm.fit)
plot(perm.fit, type="EF")

par(op)

# Logistic regression ------------------------------------------------
data(Heart)
X <- Heart$X
y <- Heart$y

op <- par(mfrow=c(2,2))
fit <- ncvreg(X, y, family="binomial")
plot(fit, main=expression(paste(gamma,"=",3)))
fit <- ncvreg(X, y, family="binomial", gamma=10)
plot(fit, main=expression(paste(gamma,"=",10)))
fit <- ncvreg(X, y, family="binomial", gamma=1.5)
plot(fit, main=expression(paste(gamma,"=",1.5)))
fit <- ncvreg(X, y, family="binomial", penalty="SCAD")
plot(fit, main=expression(paste("SCAD, ",gamma,"=",3)))

par(op)

op <- par(mfrow=c(2,2))
fit <- ncvreg(X, y, family="binomial")
plot(fit, main=expression(paste(alpha,"=",1)))
fit <- ncvreg(X, y, family="binomial", alpha=0.9)
plot(fit, main=expression(paste(alpha,"=",0.9)))
fit <- ncvreg(X, y, family="binomial", alpha=0.5)
plot(fit, main=expression(paste(alpha,"=",0.5)))
fit <- ncvreg(X, y, family="binomial", alpha=0.1)
plot(fit, main=expression(paste(alpha,"=",0.1)))

par(op)